$\,_{3}F_{4}$ hypergeometric functions as a sum of a product of $\,_{2}F_{3}$ functions

Abstract: This paper shows that certain $\,{3}F{4}$ hypergeometric functions may be expanded in sums of pair products of $\,{2}F{3}$ functions. This expands the class of hypergeometric functions having summation theorems beyond those expressible as pair-products of generalized Whittaker functions, $\,{2}F{1}$ functions, and $\,{3}F{2}$ functions into the realm of $\,{P}F{Q}$ functions where $P<Q$ for both the summand and terms in the series. In addition to its intrinsic value, this result has a specific application in calculating the response of the atoms to laser stimulation in the Strong Field Approximation.